In mathematics, and specifically in potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given...

the delta function. The Poisson kernel is also closely related to the Cauchy distribution and Epanechnikov and Gaussian kernel functions. This semigroup...

probability Poisson summation formula in Fourier analysis Poisson kernel in complex or harmonic analysis Poisson–Jensen formula in complex analysis This disambiguation...

Baron Siméon Denis Poisson (/pwɑːˈsɒ̃/, US also /ˈpwɑːsɒn/; French: [si.me.ɔ̃ də.ni pwa.sɔ̃]; 21 June 1781 – 25 April 1840) was a French mathematician...

are connected with the derivatives of the Poisson integral kernel. For each positive integer n the Poisson wavelet ψ n ( t ) {\displaystyle \psi _{n}(t)}...

two variables, that is called the kernel or nucleus of the transform. Some kernels have an associated inverse kernel K − 1 ( u , t ) {\displaystyle K^{-1}(u...

{y}{(x-s)^{2}+y^{2}}}\;\mathrm {d} s} which is the convolution of f with the Poisson kernel P ( x , y ) = y π ( x 2 + y 2 ) {\displaystyle P(x,y)={\frac {y}{\pi...

can regain a (harmonic) function f on the unit disk by means of the Poisson kernel Pr: f ( r e i θ ) = 1 2 π ∫ 0 2 π P r ( θ − ϕ ) f ~ ( e i ϕ ) d ϕ ,...

theory because it is the simplest Furstenberg measure, the classical Poisson kernel associated with a Brownian motion in a half-plane. Conjugate harmonic...

moment generating function. In mathematics, it is closely related to the Poisson kernel, which is the fundamental solution for the Laplace equation in the upper...

statistics and related fields, a Poisson point process (also known as: Poisson random measure, Poisson random point field and Poisson point field) is a type of...

and the solution to the problem (at least for the ball) using the Poisson kernel was known to Dirichlet (judging by his 1850 paper submitted to the Prussian...

}r^{|n|}e^{in\omega }} is the Poisson kernel on the unit disk. If the function f {\displaystyle f} has no zeros in the unit disk, the Poisson-Jensen formula reduces...

mathematics as well. For instance, in harmonic analysis the Poisson kernel is a reproducing kernel for the Hilbert space of square-integrable harmonic functions...

d H 1 {\displaystyle d\omega (X,\mathbb {D} )/dH^{1}} is called the Poisson kernel. More generally, if n ≥ 2 {\displaystyle n\geq 2} and B n = { X ∈ R...

{\displaystyle K(z,\xi )={\frac {1-|z|^{2}}{|\xi -z|^{2}}}} is the Poisson kernel, holds for all z ∈ D {\displaystyle z\in \mathbb {D} } . One way to...

derive the following interesting[clarification needed] identity from the Poisson summation formula: ∑ k ∈ Z exp ⁡ ( − π ⋅ ( k c ) 2 ) = c ⋅ ∑ k ∈ Z exp...

{\omega }}.} The integral can be solved by analytically continuing the Poisson kernel, giving G ( t , x ) = lim ϵ → 0 + C D D − 1 Im ⁡ [ ‖ x ‖ 2 − ( t − i...

solve Poisson's differential equation Poisson differential operator Dirichlet–Poisson problem Discrete Poisson equation Poisson kernel Poisson integral...

The result can be proven analytically, using the properties of the Poisson kernel in the unit ball, or geometrically by applying a rotation to the vector...

|t|>\delta } . The Fejér kernel The Poisson kernel (continuous index) The Landau kernel The Dirichlet kernel is not a summability kernel, since it fails the...

H^{p}(\mathbb {D} )} is the Hardy space. The proof utilizes the symmetry of the Poisson kernel using the Hardy–Littlewood maximal function for the circle. The analogous...

{\displaystyle u(x,y)=\int _{\mathbb {R} ^{n}}P_{y}(t)f(x-t)\,dt} where the Poisson kernel P on the upper half space { ( y ; x ) ∈ R n + 1 ∣ y > 0 } {\displaystyle...

Lebesgue point of f. In fact the operator T1 − εHf has kernel Qr + i, where the conjugate Poisson kernel Qr is defined by Q r ( θ ) = 2 r sin ⁡ θ 1 − 2 r cos...

potential. Similar expressions are available for the expansion of the Poisson kernel in a ball (Stein & Weiss 1971). It follows that the quantities C k (...

)(e^{i\theta })=\sup _{0\leq r<1}\varphi (re^{i\theta }).} If Pr denotes the Poisson kernel, it follows from the subharmonicity that 0 ≤ φ ( r e i θ ) ≤ 1 2 π ∫...

for some polynomial P i j {\displaystyle P_{ij}} . Hilbert Transform Poisson kernel Riesz potential Strictly speaking, the definition (1) may only make...

In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values...

_{\delta }^{1}(1-t^{2})^{n}\,dt\leq (n+1)(1-r^{2})^{n}} Poisson kernel Fejér kernel Dirichlet kernel Terras, Audrey (May 25, 2009). "Lecture 8. Dirac and...

d\mu (\theta ).} This follows from the previous theorem because: the Poisson kernel is the real part of the integrand above the real part of a holomorphic...